Dirichlet series
| $L(s,f)$ = 1 | − 0.577·3-s + 1.54·5-s − 0.318·7-s + 0.333·9-s + 0.550·11-s − 1.02·13-s − 0.892·15-s − 0.432·17-s + 1.08·19-s + 0.183·21-s − 0.205·23-s + 1.39·25-s − 0.192·27-s + 0.809·29-s − 0.819·31-s − 0.317·33-s − 0.492·35-s − 1.74·37-s + 0.591·39-s + 0.159·41-s + 0.444·43-s + 0.515·45-s − 0.354·47-s − 0.898·49-s + 0.249·51-s + 1.16·53-s + 0.851·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s,f)=\mathstrut & 24 ^{s/2} \, \Gamma_{\R}(s+1.91i) \, \Gamma_{\R}(s-1.91i) \, L(s,f)\cr =\mathstrut & \, \Lambda(1-s,f) \end{aligned}\]
Invariants
| Degree: | \(2\) |
| Conductor: | \(24\) = \(2^{3} \cdot 3\) |
| Sign: | $1$ |
| Arithmetic: | no |
| Primitive: | yes |
| Self-dual: | yes |
| Selberg data: | \((2,\ 24,\ (1.91464670089i, -1.91464670089i:\ ),\ 1)\) |
Euler product
\(L(s,f) = \displaystyle\prod_{p\ \mathrm{bad}} (1- a(p) p^{-s})^{-1} \prod_{p\ \mathrm{good}} (1- a(p) p^{-s} + \chi(p)p^{-2s})^{-1}\)
Imaginary part of the first few zeros on the critical line